You will get a detailed personal kit list when you book. This is the overview. The principle is layers: you will go from humid rainforest to sub-zero summit night in the same trip, and you manage that by adding and removing thin layers rather than carrying one heavy coat.
Waterproof hiking boots with ankle support, broken in well before you travel. Warm hiking socks (several pairs), plus camp shoes or trainers for the evenings. Gaiters are useful on the scree.
High-factor sunscreen and SPF lip balm, sunglasses (category 3–4), personal first-aid and blister kit, any prescription medication, water bottles and/or a hydration bladder (3 L total), water-purification tablets, wet wipes, and a power bank. Snacks you like — energy bars, nuts, sweets.
Down jackets, sleeping bags, poles, duffels and gaiters can all be hired in Moshi or Arusha in good condition, which is worth it if you do not climb often. Boots and anything worn next to the skin you should bring yourself.